Have you ever had a student look at you like you're talking in a foreign language during a math lesson? It happens. Sometimes there are gaps in students' prior knowledge. Sometimes students need more time to process a new concept in order to construct and develop their understanding. Sometimes I haven't explained something clearly and I need to think of a new approach to tackle this topic. When I encounter those blank stares, I think, "That's Numberwang."

Numberwang is a skit from That Mitchell and Webb Look, a British sketch comedy show from 2006. If you've never seen it, take a two minutes to watch a video of this skit. The premise of the skit is that while the presenter and contestants seem to understand the rules perfectly, they are completely inscrutable to the viewer. We're left scratching our heads in confusion just like our students sometimes do in class.

Delve!

So what do you do when you sense that students in your class are not getting it? I suggest that you invest some time to uncover your students' thinking. As Guildenstern implores in Tom Stoppard's play Rosencrantz and Guildenstern Are Dead, "Delve. Probe the background, establish the situation." Take the opportunity check in with students to determine their level of understanding. Here are some strategies you might use:

Student Self-Assessment: Some teachers periodically ask students to use hand signals to express their understanding and confidence during a lesson. This strategy can let you quickly check in with a large group of students. There are numerous other strategies to get quick student feedback during or after a lesson. A strategy for a more detailed response with higher quality feedback might use a single point rubric.

Exit Tickets: Take the last few minutes of class to have students respond to a short question. The question might be a quick math problem or a reflection on their learning from that day. Students hand in their response on the way out the door. Andrew Crandall uses exit tickets often to plan for future lessons in response to common misconceptions.

My Favourite No: A routine that can be used any time is called "My Favourite No" demonstrated by Leah Alcala. She asks her students to answer a single question on a note card and pass it in. She then quickly sorts the cards into two piles. One pile of correct answers and one pile of incorrect answers. She then goes through the incorrect answers and selects one as her favourite that displays a common error and then talks through the misconception and how to correct it. This strategy results in immediate feedback to students and a positive message that we all learn from our mistakes.

Reflect and Respond

Once you have a better picture of the misunderstandings and misconceptions that may be present in your class, you can plan your next steps. Was there really a misunderstanding or did you make assumptions about prior knowledge that weren't true? Were just a few students struggling or was it a commonly held misconception? Tracy Zager, in her book Becoming the Math Teacher You Wish You'd Had, writes, "If just a few students were confused, she could work with them individually. If there was a really interesting mistake, or patterns among the misunderstandings she saw, she could use those examples as her next teaching opportunity."

When I see those "Numberwang" looks I am reminded that even a well planned lesson can sometimes miss the mark. Reflecting on how a lesson went and how I can improve it helps me refine my teaching practice and be more responsive to students' needs. Don't let those "Numberwang" moments go by ignored. Matt Larson, in his August NCTM president's message, wrote "Making mistakes, getting feedback from our colleagues, and making iterative improvement are part of the natural process of continual growth. We should never forget that perseverance isn't just for students—perseverance also applies to us as professionals."

My nephew visited me this summer during a family vacation. He will soon be starting grade 11 (junior) ​year. He is a hard working student and would like to study mathematics in university. He understands that this is the year when he needs to start thinking about university and scholarship applications. We started chatting about his post-secondary plans one evening and he asked me two questions. The first question was, "What can I do to make sure I have competitive university and scholarship applications?" The second question was, "What sorts of careers are possible with a mathematics degree?" Here is the advice that I offered him.

Break Out from the Pack

As a high school teacher, I have written a lot of reference letters. The hardest reference letters to write are for talented students who come to class every day and work hard but have nothing to set them apart from the other 30 or so students in class. Ask yourself, "What sets me apart or makes me different and special?" Here are some things I suggested to my nephew:

The first piece of advice is simple but needs to be said... go to class every day. If you are not going to be in class, let your teacher know in advance. Similarly, pass in every assignment/project on time. If something is going to be late, let your teacher know in advance. This is what being mature and responsible looks like in high school.

Spend some time this summer doing the Brilliant.org 100 Day Summer Challenge. If you haven't been doing these problems, get started now before the summer is over. Brilliant.org posts an interesting math problem each day. Take a few minutes each morning to try the day's problem and then read through the discussion of solutions afterwards. Make note of your favourite problems and share a few of them with your teacher. One of my favourites so far was day 68, The Desert Trek.

Let your teacher know that you'd like to participate in upcoming mathematics contests and competitions. For my nephew this would be the AMC 12 math contest in February. He might also consider taking Art of Problem Solving's AMC 12 Preparation course. It meets online in the evening once a week for 12 weeks. A course like this can introduce you to interesting problem solving techniques that you might not have seen before.

These are the sorts of things that will let your teacher know that you are really interested in mathematics and points that can be included in a reference letter. Plus, if you like math, you will really enjoy the challenge.

Careers for Mathematians

There are some great careers for mathematicians that don't get a lot of press. Here are some careers that I find really interesting:

Data Scientist - Data science is all about extracting knowledge and gaining insights from data. Hal Varian, Google's Chief Economist, once said, "The ability to take data – to be able to understand it, to process it, to extract value from it, to visualize it, to communicate it’s going to be a hugely important skill in the next decades." He went on to say that there are enormous quantities of free data but limited capabilities and resources to understand that data and extract value from it.

Forensic Accountant - This career is not nearly as dangerous as the 2016 movie The Accountant starring Ben Affleck might lead you to believe. Forensic accountants investigate fraud or embezzlement and analyze financial information for use in legal proceedings.

Actuarial Scientist - Actuaries manage risk. They use analytical and statistical skills to help organizations plan for the future and protect themselves from loss. This career is often listed as one of the top-ranked jobs based on a variety of factors including salary and career outlook. Check out this Taking Maths Further podcast which includes an interview with Simon Perera from Lane, Clark & Peacock about his work as an actuary, what an actuary is and how it involves predicting the growth of investments.

So what advice would you give to a grade 11 student who is interested in a career in mathematics? I'd love to know what you think.